Local Search for Vehicle Routing and Scheduling Problems: Review and Conceptual Integration

Local search and local search-based metaheuristics are currently the only available methods for obtaining good solutions to large vehicle routing and scheduling problems. In this paper we provide a review of both classical and modern local search neighborhoods for this class of problems. The intention of this paper is not only to give an overview but to classify and analyze the structure of different neighborhoods. The analysis is based on a formal representation of VRSP solutions given by a unifying giant-tour model. We describe neighborhoods implicitly by a set of transformations called moves and show how moves can be decomposed further into partial moves. The search method has to compose these partial moves into a complete move in an efficient way. The goal is to find a local best neighbor and to reach a local optimum as quickly as possible. This can be achieved by search methods, which do not scan all neighbor solutions explicitly. Our analysis shows how the properties of the partial moves and the constraints of the VRSP influences the choice of an appropriate search technique.     

Creating very large scale neighborhoods out of smaller ones by compounding moves

This paper discusses neighborhood search algorithms where the size of the neighborhood is “very large” with respect to the size of the input data. We concentrate on such a very large scale neighborhood (VLSN) search technique based on compounding independent moves (CIM) such as 2-opts, swaps, and insertions. We present a systematic way of creating and searching CIM neighborhoods for routing problems with side constraints. For such problems, the exact search of the CIM neighborhood becomes NP-hard. We introduce a multi-label shortest path algorithm for searching these neighborhoods heuristically. Results of a computational study on the vehicle routing problem with capacity and distance restrictions shows that CIM algorithms are very competitive approaches for solving vehicle routing problems. Overall, the solutions generated by the CIM algorithm have the best performance among the current solution methodologies in terms of percentage deviation from the best-known solutions for large-scale capacitated VRP instances.     

An efficient heuristic algorithm for the capacitated p-\! median problem

The capacitated $p$ -median problem (CPMP) is one of the well-known facility-location problems. The objective of the problem is to minimize total cost of locating a set of capacitated service points and allocating a set of demand points to the located service points, while the total allocated demand for each service point is not be greater than its capacity limit. This paper presents an efficient heuristic algorithm based on the local branching and relaxation induced neighborhood search methods for the CPMP. The proposed algorithm is a heuristic technique that utilizes a general mixed integer programming solver to explore neighborhoods. The parameters of the proposed algorithm are tuned by design of experiments. The proposed method is tested on a large set of benchmark instances. The results show that the method outperforms the best method found in the literature.     

Local search algorithms for the rectangle packing problem with general spatial costs

We propose local search algorithms for the rectangle packing problem to minimize a general spatial cost associated with the locations of rectangles. The problem is to pack given rectangles without overlap in the plane so that the maximum cost of the rectangles is minimized. Each rectangle has a set of modes, where each mode specifies the width and height of the rectangle and its spatial cost function. The spatial costs are general piecewise linear functions which can be non-convex and discontinuous. Various types of packing problems and scheduling problems can be formulated in this form. To represent a solution of this problem, a pair of permutations of n rectangles is used to determine their horizontal and vertical partial orders, respectively. We show that, under the constraint specified by such a pair of permutations, optimal locations of the rectangles can be efficiently determined by using dynamic programming. The search for finding good pairs of permutations is conducted by local search and metaheuristic algorithms. We report computational results on various implementations using different neighborhoods, and compare their performance. We also compare our algorithms with other existing heuristic algorithms for the rectangle packing problem and scheduling problem. These computational results exhibit good prospects of the proposed algorithms. Key words.rectangle packing – sequence pair – general spatial cost – dynamic programming – metaheuristicsMathematics Subject Classification (1991):20E28, 20G40, 20C20     

A measure concentration inequality for contracting markov chains

The concentration of measure phenomenon in product spaces means the following: if a subsetA of then'th power of a probability space does not have too small a probability then very large probability is concentrated in a small neighborhood ofA. The neighborhood is in many cases understood in the sense of Hamming distance, and then measure concentration is known to occur for product probability measures, and also for the distribution of some processes with very fast and uniform decay of memory. Recently Talagrand introduced another notion of neighborhood of sets for which he proved a similar measure concentration inequality which in many cases allows more efficient applications than the one for a Hamming neighborhood. So far this inequality has only been proved for product distributions. The aim of this paper is to give a new proof of Talagrand's inequality, which admits an extension to contracting Markov chains. The proof is based on a new asymmetric notion of distance between probability measures, and bounding this distance by informational divergence. As an application, we analyze the bin packing problem for Markov chains.This work was supported by the grants OTKA 1906 and T 016386 of the Hungarian Academy of Sciences, and by MTA-NSF project 37.     

Neighborhood analysis: a case study on curriculum-based course timetabling

In this paper, we present an in-depth analysis of neighborhood relations for local search algorithms. Using a curriculum-based course timetabling problem as a case study, we investigate the search capability of four neighborhoods based on three evaluation criteria: percentage of improving neighbors, improvement strength and search steps. This analysis shows clear correlations of the search performance of a neighborhood with these criteria and provides useful insights on the very nature of the neighborhood. This study helps understand why a neighborhood performs better than another one and why and how some neighborhoods can be favorably combined to increase their search power. This study reduces the existing gap between reporting experimental assessments of local search-based algorithms and understanding their behaviors.     

A general variable neighborhood search for solving the uncapacitated single allocation p-hub median problem

We present a new general variable neighborhood search approach for the uncapacitated single allocation p-hub median problem in networks. This NP hard problem is concerned with locating hub facilities in order to minimize the traffic between all origin-destination pairs. We use three neighborhoods and efficiently update data structures for calculating new total flow in the network. In addition to the usual sequential strategy, a new nested strategy is proposed in designing a deterministic variable neighborhood descent local search. Our experimentation shows that general variable neighborhood search based heuristics outperform the best-known heuristics in terms of solution quality and computational effort. Moreover, we improve the best-known objective values for some large Australia Post and PlanetLab instances. Results with the new nested variable neighborhood descent show the best performance in solving very large test instances.     

TSPACK: A two-level tabu search for the three-dimensional bin packing problem

Three-dimensional orthogonal bin packing is a problem NP-hard in the strong sense where a set of boxes must be orthogonally packed into the minimum number of three-dimensional bins. We present a two-level tabu search for this problem. The first-level aims to reduce the number of bins. The second optimizes the packing of the bins. This latter procedure is based on the Interval Graph representation of the packing, proposed by Fekete and Schepers, which reduces the size of the search space. We also introduce a general method to increase the size of the associated neighborhoods, and thus the quality of the search, without increasing the overall complexity of the algorithm. Extensive computational results on benchmark problem instances show the effectiveness of the proposed approach, obtaining better results compared to the existing ones.     

New relationships for multi-neighborhood search for the minimum linear arrangement problem

We develop a series of theorems about the graph structure of the classical Minimum Linear Arrangement (MinLA) problem which disclose properties that can be exploited by Multi-Neighborhood Search (MNS) algorithms. As a foundation, we differentiate between swaps of labels attached to adjacent and non-adjacent nodes to create two new neighborhood classes, and show how our theorems yield efficient algorithms for updating key arrays used by local search procedures. In addition, we introduce a class of neighborhoods called set-based neighborhoods supported by a theorem that identifies solutions (labelings) for the MinLA problem in polynomial time that dominate exponential numbers of alternative solutions. The component neighborhoods within this new neighborhood class can be applied in various sequences in conjunction with the first two new neighborhoods introduced. Our results also apply to problems with objectives different than those of MinLA. Finally, our results make it possible to exploit the new neighborhoods according to the user's choice of MNS protocols and alternative local search algorithms.     

A general approach for optimizing regular criteria in the job-shop scheduling problem

Even though a very large number of solution methods has been developed for the job-shop scheduling problem, a majority has been designed for the makespan criterion. In this paper, we propose a general approach for optimizing any regular criterion in the job-shop scheduling problem. The approach is a local search method that uses a disjunctive graph model and neighborhoods generated by swapping critical arcs. The connectivity property of the neighborhood structure is proved and a novel efficient method for evaluating moves is presented. Besides its generality, another prominent advantage of the proposed approach is its simple implementation that only requires to tune the range of one parameter. Extensive computational experiments carried out on various criteria (makespan, total weighted flow time, total weighted tardiness, weighted sum of tardy jobs, maximum tardiness) show the efficiency of the proposed approach. Best results were obtained for some problem instances taken from the literature.     

Extended neighborhood: Definition and characterization

We consider neighborhood search defined on combinatorial optimization problems. Suppose that N is a Neighborhood for combinatorial optimization problem X. We say that N is LO-equivalent (locally optimal) to N if for any instance of X, the set of locally optimal solutions with respect to N and N are the same. The union of two LO-equivalent neighborhoods is itself LO-equivalent to the neighborhoods. The largest neighborhood that is LO-equivalent to N is called the extended neighborhood of N, and denoted as N^*. We analyze some basic properties of the extended neighborhood. We provide a geometric characterization of the extended neighborhood N^* when the instances have linear costs defined over a cone. For the TSP, we consider 2-opt^*, the extended neighborhood for the 2-opt (i.e., 2-exchange) neighborhood structure. We show that number of neighbors of each tour T in 2-opt^* is at least (n /2 -2)!. We show that finding the best tour in the 2-opt^* neighborhood is NP-hard. We also show that the extended neighborhood for the graph partition problem is the same as the original neighborhood, regardless of the neighborhood defined. This result extends to the quadratic assignment problem as well. This result on extended neighborhoods relies on a proof that the convex hull of solutions for the graph partition problem has a diameter of 1, that is, every two corner points of this polytope are adjacent.Acknowledgement We thank Ravi Ahuja, Andreas Schulz, and Ozlem Ergun for useful discussions. This research was supported through NSF contract DMI-9820998.     

Using Constraint-Based Operators to Solve the Vehicle Routing Problem with Time Windows

This paper presents operators searching large neighborhoods in order to solve the vehicle routing problem. They make use of the pruning and propagation techniques of constraint programming which allow an efficient search of such neighborhoods. The advantages of using a large neighborhood are not only the increased probability of finding a better solution at each iteration but also the reduction of the need to invoke specially-designed methods to avoid local minima. These operators are combined in a variable neighborhood descent in order to take advantage of the different neighborhood structures they generate.     

A geneal vaiable neighbohood seach fo solving the uncapacitated $$$$-allocation $$p$$p-hub median poblem

The \(p\)-hub median problem consists of choosing \(p\) hub locations from a set of nodes with pairwise traffic demands in order to route the traffic between the origin-destination pairs at minimum cost. We accept general assumption that transportation between non-hub nodes is possible only via \(r\)-hub nodes, to which non-hub nodes are assigned. In this paper we propose a general variable neighborhood search heuristic to solve the problem in an efficient and effective way. Moreover, for the first time full nested variable neighborhood descent is applied as a local search within Variable neighborhood search. Computational results outperform the current state-of-the-art results obtained by GRASP based heuristic.     

A hybrid algorithm for constrained order packing

Constraint order packing, which is an extension to the classical two-dimensional bin packing, adds an additional layer of complexity to known bin packing problems by new additional placement and order constraints. While existing meta heuristics usually produce good results for common bin packing problems in any dimension, they are not able to take advantage of special structures resulting from these constraints in this particular two-dimensional prolbem type. We introduce a hybrid algorithm that is based on greedy search and is nested within a network search algorithm with dynamic node expansion and meta logic, inspired by human intuition, to overrule decisions implied by the greedy search. Due to the design of this algorithm we can control the performance characteristics to lie anywhere between classical network search algorithms and local greedy search. We will present the algorithm, discuss bounds and show that their performance outperforms common approaches on a variety of data sets based on industrial applications. Furthermore, we discuss time complexity and show some ideas to speed up calculations and improve the quality of results.     

Bin packing with general cost structures

Following the work of Anily et?al., we consider a variant of bin packing called bin packing with general cost structures (GCBP) and design an asymptotic fully polynomial time approximation scheme (AFPTAS) for this problem. In the classic bin packing problem, a set of one-dimensional items is to be assigned to subsets of total size at most 1, that is, to be packed into unit sized bins. However, in GCBP, the cost of a bin is not 1 as in classic bin packing, but it is a non-decreasing and concave function of the number of items packed in it, where the cost of an empty bin is zero. The construction of the AFPTAS requires novel techniques for dealing with small items, which are developed in this work. In addition, we develop a fast approximation algorithm which acts identically for all non-decreasing and concave functions, and has an asymptotic approximation ratio of 1.5 for all functions simultaneously.     

Matching based very large-scale neighborhoods for?parallel machine scheduling

In this paper we study very large-scale neighborhoods for the minimum total weighted completion time problem on parallel machines, which is known to be strongly $\mathcal{NP}$ -hard. We develop two different ideas leading to very large-scale neighborhoods in which the best improving neighbor can be determined by calculating a weighted matching. The first neighborhood is introduced in a general fashion using combined operations of a basic neighborhood. Several examples for basic neighborhoods are given. The second approach is based on a partitioning of the job sets on the machines and a reassignment of them. In a computational study we evaluate the possibilities and the limitations of the presented very large-scale neighborhoods.     

On characterization of maximal independent sets via quadratic optimization

This article investigates the local maxima properties of a box-constrained quadratic optimization formulation of the maximum independent set problem in graphs. Theoretical results characterizing binary local maxima in terms of certain induced subgraphs of the given graph are developed. We also consider relations between continuous local maxima of the quadratic formulation and binary local maxima in the Hamming distance-1 and distance-2 neighborhoods. These results are then used to develop an efficient local search algorithm that provides considerable speed-up over a typical local search algorithm for the binary Hamming distance-2 neighborhood.     

A “Logic-Constrained” Knapsack Formulation and a Tabu Algorithm for the Daily Photograph Scheduling of an Earth Observation Satellite

The daily photograph scheduling problem of earth observation satellites such as Spot 5 consists of scheduling a subset of mono or stereo photographs from a given set of candidates to different cameras. The scheduling must maximize a profit function while satisfying a large number of constraints. In this paper, we first present a formulation of the problem as a generalized version of the well-known knapsack model, which includes large numbers of binary and ternary logical constraints. We then develop a tabu search algorithm which integrates some important features including an efficient neighborhood, a dynamic tabu tenure mechanism, techniques for constraint handling, intensification and diversification. Extensive experiments on a set of large and realistic benchmark instances show the effectiveness of this approach.     

A filter-and-fan approach to the job shop scheduling problem

The job shop scheduling problem (JSSP) is a notoriously difficult problem in combinatorial optimization. Extensive investigation has been devoted to developing efficient algorithms to find optimal or near-optimal solutions. This paper proposes a new heuristic algorithm for the JSSP that effectively combines the classical shifting bottleneck procedure (SBP) with a dynamic and adaptive neighborhood search procedure. Our new search method, based on a filter-and-fan (F&F) procedure, uses the SBP as a subroutine to generate a starting solution and to enhance the best schedules produced. The F&F approach is a local search procedure that generates compound moves by a strategically abbreviated form of tree search. Computational results carried out on a standard set of 43 benchmark problems show that our F&F algorithm performs more robustly and effectively than a number of leading metaheuristic algorithms and rivals the best of these algorithms.